623 research outputs found
A quantitative probabilistic investigation into the accumulation of rounding errors in numerical ODE solution.
We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs) from a probabilistic viewpoint. We show that the accumulation of rounding errors results in a solution which is inherently random and we obtain the theoretical distribution of the trajectory as a function of time, the step size and the numerical precision of the computer. We consider, in particular, systems which amplify the effect of the rounding errors so that over long time periods the solutions exhibit divergent behaviour. By performing multiple repetitions with different values of the time step size, we observe numerically the random distributions predicted theoretically. We mainly focus on the explicit Euler and fourth order Runge–Kutta methods but also briefly consider more complex algorithms such as the implicit solvers VODE and RADAU5 in order to demonstrate that the observed effects are not specific to a particular method
Nekhoroshev theorem for the periodic Toda lattice
The periodic Toda lattice with sites is globally symplectomorphic to a
two parameter family of coupled harmonic oscillators. The action
variables fill out the whole positive quadrant of . We prove that in
the interior of the positive quadrant as well as in a neighborhood of the
origin, the Toda Hamiltonian is strictly convex and therefore Nekhoroshev's
theorem applies on (almost) all parts of phase space.Comment: 28 page
Some critical remarks on a new numerical method for simulation of dynamical systems
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/68690/2/10.1177_003754976600600210.pd
Global Birkhoff coordinates for the periodic Toda lattice
In this paper we prove that the periodic Toda lattice admits globally defined
Birkhoff coordinates.Comment: 32 page
Texture and shape of two-dimensional domains of nematic liquid crystal
We present a generalized approach to compute the shape and internal structure
of two-dimensional nematic domains. By using conformal mappings, we are able to
compute the director field for a given domain shape that we choose from a rich
class, which includes drops with large and small aspect ratios, and sharp
domain tips as well as smooth ones. Results are assembled in a phase diagram
that for given domain size, surface tension, anchoring strength, and elastic
constant shows the transitions from a homogeneous to a bipolar director field,
from circular to elongated droplets, and from sharp to smooth domain tips. We
find a previously unaccounted regime, where the drop is nearly circular, the
director field bipolar and the tip rounded. We also find that bicircular
director fields, with foci that lie outside the domain, provide a remarkably
accurate description of the optimal director field for a large range of values
of the various shape parameters.Comment: 12 pages, 10 figure
The structure of atomic nitrogen adsorbed on Fe(100)
Nitrogen atoms adsorbed on a Fe(100) surface cause the formation of an ordered c(2 × 2) overlayer with coverage 0.5. A structure analysis was performed by comparing experimental LEED I–V spectra with the results of multiple scattering model calculations. The N atoms were found to occupy fourfold hollow sites, with their plane 0.27 Å above the plane of the surface Fe atoms. In addition, nitrogen adsorption causes an expansion of the two topmost Fe layers by 10% (= 0.14 Å). The minimum r-factor for this structure analysis is about 0.2 for a total of 16 beams. The resulting atomic arrangement is similar to that in the (002) plane of bulk Fe4N, thus supporting the view of a “surface nitride” and providing a consistent picture of the structural and bonding properties of this surface phase
Simple and Nearly Optimal Polynomial Root-finding by Means of Root Radii Approximation
We propose a new simple but nearly optimal algorithm for the approximation of
all sufficiently well isolated complex roots and root clusters of a univariate
polynomial. Quite typically the known root-finders at first compute some crude
but reasonably good approximations to well-conditioned roots (that is, those
isolated from the other roots) and then refine the approximations very fast, by
using Boolean time which is nearly optimal, up to a polylogarithmic factor. By
combining and extending some old root-finding techniques, the geometry of the
complex plane, and randomized parametrization, we accelerate the initial stage
of obtaining crude to all well-conditioned simple and multiple roots as well as
isolated root clusters. Our algorithm performs this stage at a Boolean cost
dominated by the nearly optimal cost of subsequent refinement of these
approximations, which we can perform concurrently, with minimum processor
communication and synchronization. Our techniques are quite simple and
elementary; their power and application range may increase in their combination
with the known efficient root-finding methods.Comment: 12 pages, 1 figur
Технологии производства валков и последние исследования по изготовлению валков для станов холодной прокатки
Pioneer 10 Doppler data analysis: disentangling periodic and secular anomalies
This paper reports the results of an analysis of the Doppler tracking data of
Pioneer probes which did show an anomalous behaviour. A software has been
developed for the sake of performing a data analysis as independent as possible
from that of J. Anderson et al. \citep{anderson}, using the same data set. A
first output of this new analysis is a confirmation of the existence of a
secular anomaly with an amplitude about 0.8 nms compatible with that
reported by Anderson et al. A second output is the study of periodic variations
of the anomaly, which we characterize as functions of the azimuthal angle
defined by the directions Sun-Earth Antenna and Sun-Pioneer. An
improved fit is obtained with periodic variations written as the sum of a
secular acceleration and two sinusoids of the angles and .
The tests which have been performed for assessing the robustness of these
results are presented.Comment: 13 pages, 6 figures, minor amendment
On Kaluza's sign criterion for reciprocal power series
T. Kaluza has given a criterion for the signs of the power series of a
function that is the reciprocal of another power series. In this note the
sharpness of this condition is explored and various examples in terms of the
Gaussian hypergeometric series are given. A criterion for the monotonicity of
the quotient of two power series due to M. Biernacki and J. Krzy\.z is applied.Comment: 13 page
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